In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an invariant measure is a measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 that is preserved by some function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. Ergodic theory

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

 is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.

Definition

Let (X, Σ) be a measurable space and let f be a measurable function

Measurable function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

 from X to itself. A measure μ on (X, Σ) is said to be invariant under f if, for every measurable set A in Σ,


In terms of the push forward

Pushforward measure

In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...

, this states that f_∗(μ) = μ.

The collection of measures (usually probability measure

Probability measure

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...

s) on X that are invariant under f is sometimes denoted M_f(X). The collection of ergodic measures

Ergodic (adjective)

In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.-Etymology:The word ergodic is...

, E_f(X), is a subset of M_f(X). Moreover, any convex combination

Convex combination

In convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1....

 of two invariant measures is also invariant, so M_f(X) is a convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

; E_f(X) consists precisely of the extreme points of M_f(X).

In the case of a dynamical system

Dynamical system (definition)

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space...

 (X, T, φ), where (X, Σ) is a measurable space as before, T is a monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

 and φ : T × X → X is the flow map, a measure μ on (X, Σ) is said to be an invariant measure if it is an invariant measure for each map φ_t : X → X. Explicitly, μ is invariant if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

Put another way, μ is an invariant measure for a sequence of random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s (Z_t)_t≥0 (perhaps a Markov chain

Markov chain

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...

 or the solution to a stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

) if, whenever the initial condition Z₀ is distributed according to μ, so is Z_t for any later time t.

Examples

• Consider the real line
  Real line
  In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
  
   R with its usual Borel σ-algebra; fix a ∈ R and consider the translation map T_a : R → R given by:

Then one-dimensional Lebesgue measure

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

 λ is an invariant measure for T_a.

• More generally, on n-dimensional Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   Rⁿ with its usual Borel σ-algebra, n-dimensional Lebesgue measure λⁿ is an invariant measure for any isometry
  Isometry
  In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
  
   of Euclidean space, i.e. a map T : Rⁿ → Rⁿ that can be written as

for some n × n orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

A ∈ O(n) and a vector b ∈ Rⁿ.

• The invariant measure in the first example is unique up to trivial renormalization with a constant factor. This does not have to be necessarily the case: Consider a set consisting of just two points and the identity map which leaves each point fixed. Then any probability measure is invariant. Note that S trivially has a decomposition into T-invariant components {A} and {B}.